We introduce and study two new classes of unital quantum channels. The first class describes a 2-parameter family of channels given by completely positive (CP) maps $M_3({\bf C}) \mapsto M_3({\bf C})$ which are both unital and trace-preserving. Almost every member of this family is factorizable and extreme in the set of CP maps which are both unital and trace-preserving, but is not extreme in either the set of unital CP maps or the set of trace-preserving CP maps.
We also study a large class of maps which generalize the Werner-Holevo channel for $d = 3$ in the sense that they are defined in terms of partial isometries of rank $d-1$. Moreover, we extend this to maps whose Kraus operators have the form $t |e_j \rangle \langle e_j | \oplus V $ with $V \in M_{d-1} ({\bf C}) $ unitary and $t \in (-1,1)$. We show that almost every map in this class is extreme in both the set of unital CP maps and the set of trace-preserving CP maps. We analyze in detail a particularly interesting subclass which is extreme unless $t = -1/(d-1)$. For $d = 3$, this includes a pair of channels which have a dual factorization in the sense that they can be obtained by taking the partial trace over different subspaces after using the same unitary conjugation in $M_3({\bf C}) \otimes M_3({\bf C})$.
Two notions for linear maps (operational convex combinations and operational exetreme points) are introduced. The set S of ucp maps on the n times n matrix algebra is the operational convex combinations of the identity map. An operational extreme point of S is an extreme point of S but the converse does not hold, and every automorphism is an operational extreme point of S.
We study the quantum ($C^*$) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, $C^*$-extreme points of normalized POVMs on countable spaces (in particular for finite sets) are always spectral measures (normalized projection valued measures). More generally it is shown that atomic $C^*$-extreme points are spectral. A Krein-Milman type theorem for POVMs has also been proved. As an application it is shown that a map on any commutative unital $C^*$-algebra with countable …
In this paper we study the $C^*$-convex set of unital entanglement breaking (EB-)maps on matrix algebras. General properties and an abstract characterization of $C^*$-extreme points are discussed. By establishing a Radon-Nikodym type theorem for a class of EB-maps we give a complete description of the $C^*$-extreme points. It is shown that a unital EB-map $Φ:M_{d_1}\to M_{d_2}$ is $C^*$-extreme if and only if it has Choi-rank equal to $d_2$. Finally, as a direct consequence of the Holevo form of EB-maps, we derive a noncommutative analogue of the Krein-Milman theorem for $C^*$-convexity of the…
We consider the convex set of ( unital ) positive ( completely ) maps from a $C^*$ algebra $\cla$ to a von-Neumann sub-algebra $\clm$ of $\clb(\clh)$, the algebra of bounded linear operators on a Hilbert space $\clh$ and study its extreme points via its canonical lifting to the convex set of ( unital ) positive ( complete ) normal maps from $\hat{\cla}$ to $\clm$, where $\hat{\cla}$ is the universal enveloping von-Neumann algebra over $\cla$. If $\cla=\clm$ and a ( complete ) positive operator $τ$ is a unique sum of a normal and a singular ( complete ) positive maps. Furthermore, a unital comp…
Let A be a unital separable simple infinite-dimensional nuclear C*-algebra with at least one tracial state. We prove that if the trace space of A has compact finite-dimensional extreme boundary then there exist unital embeddings of matrix algebras into a certain central sequence algebra of A which is determined by the uniform topology on the trace space. As an application, it is shown that if furthermore A has strict comparison then A absorbs the Jiang-Su algebra tensorially.
The generalized state space $ S_{\mathcal{H}}(\mathcal{\mathcal{A}})$ of all unital completely positive (UCP) maps on a unital $C^*$-algebra $\mathcal{A}$ taking values in the algebra $\mathcal{B}(\mathcal{H})$ of all bounded operators on a Hilbert space $\mathcal{H}$, is a $C^\ast$-convex set. In this paper, we establish a connection between $C^\ast$-extreme points of $S_{\mathcal{H}}(\mathcal{A})$ and a factorization property of certain algebras associated to the UCP map. In particular, this factorization property of some nest algebras is used to give a complete characterization of those $C^…
We show that the space of tracial quantum symmetric states of an arbitrary unital C*-algebra is a Choquet simplex and is a face of the tracial state space of the universal unital C*-algebra free product of A with itself infinitely many times. We also show that the extreme points of this simplex are dense, making it the Poulsen simplex when A is separable and nontrivial. In the course of the proof we characterize the centers of certain tracial amalgamated free product C*-algebras.
Mean dimension for AH-algebras is introduced. It is shown that if a simple unital AH-algebra with diagonal maps has mean dimension zero, then it has strict comparison on positive elements. In particular, the strict order on projections is determined by traces. Moreover, a lower bound of the mean dimension is given in term of comparison radius. Using classification results, if a simple unital AH-algebra with diagonal maps has mean dimension zero, it must be an AH-algebra without dimension growth. Two classes of AH-algebras are shown to have mean dimension zero: the class of AH-algebras with at …
We study new classes of linear preservers between C$^*$-algebras and JB$^*$-triples. Let $E$ and $F$ be JB$^*$-triples with $\partial_{e} (E_1)$. We prove that every linear map $T:E\to F$ strongly preserving Brown-Pedersen quasi-invertible elements is a triple homomorphism. Among the consequences, we establish that, given two unital C$^*$-algebras $A$ and $B,$ for each linear map $T$ strongly preserving Brown-Pedersen quasi-invertible elements, then there exists a Jordan $^*$-homomorphism $S: A\to B$ satisfying $T(x) = T(1) S(x)$, for every $x\in A$. We also study the connections between linea…
Let $M$ be a unital JB$^*$-algebra whose closed unit ball is denoted by $\mathcal{B}_M$. Let $\partial_e(\mathcal{B}_M)$ denote the set of all extreme points of $\mathcal{B}_M$. We prove that an element $u\in \partial_e(\mathcal{B}_M)$ is a unitary if and only if the set $$\mathcal{M}_{u} = \{e\in \partial_e(\mathcal{B}_M) : \|u\pm e\|\leq \sqrt{2} \}$$ contains an isolated point. This is a new geometric characterisation of unitaries in $M$ in terms of the set of extreme points of $\mathcal{B}_M$.
For A a separable unital C*-algebra and M a separable McDuff II_1-factor, we show that the space Hom_w(A,M) of weak approximate unitary equivalence classes of unital *-homomorphisms A \rightarrow M may be considered as a closed, bounded, convex subset of a separable Banach space -- a variation on N. Brown's convex structure Hom(N,R^U). When A is nuclear, Hom_w(A,M) is affinely homeomorphic to the trace space of A, but in general Hom_w(A,M) and the trace space of A do not share the same data (several examples are provided). We characterize extreme points of Hom_w(A,M) in the case where either A…
\noindent Let $A$ be an arbitrary $C^*$ algebra. In \cite{BH} Blackadar and Handelman conjectured the set of lower semicontinuous dimension functions on $A$ to be pointwise dense in the set $DF(A)$ of all dimension functions on $A$ and $DF(A)$ to be a Choquet simplex. We provide an equivalent condition for the first conjecture for unital $A$. Then by applying this condition we confirm the first Conjecture for all unital $A$ for which either the radius of comparison is finite or the semigroup $W(A)$ is almost unperforated. As far as we know the most general results on the first Conjecture up to…
We study Z-actions on unital simple separable stably finite C*-algebras of finite nuclear dimension. Assuming that the extreme boundary of the trace space is compact and finite dimensional, and that the induced action on the trace space is trivial, we show that strongly outer Z-actions have finite Rokhlin dimension in the sense of Hirshberg, Winter and Zacharias.
We introduce symmetric states and quantum symmetric states on universal unital free product C*-algebras an arbitrary unital C*-algebra A with itself infinitely many times, as a generalization of the notions of exchangeable and quantum exchangeable random variables. We prove existence of conditional expectations onto tail algebras in various settings and we define a natural C*-subalgebra of the tail algebra, called the tail C*-algebra. Extending and building on the proof of the noncommutative de Finetti theorem of Koestler and Speicher, we prove a de Finetti type theorem that characterizes quan…
Let $A$ be a unital C$^*$-algebra and let $σ$ be a one-parameter automorphism group of $A$. We consider $\operatorname{QSS}_σ(A)$, the set of all quantum symmetric states on $*_1^\infty A$ that are also KMS states (for a fixed inverse temperature, for specificity taken to be $-1$) for the free product automorphism group $*_1^\inftyσ$. We characterize the elements of $\operatorname{QSS}_σ(A)$, we show that $\operatorname{QSS}_σ(A)$ is a Choquet simplex whenever it is nonempty and we characterize its extreme points.
